How to convert from NetworkX graph to ete3 Tree object?

I'm trying to figure out how to build an ete3.Tree object from a networkx directed graph? I added each child in the way I thought would produce the desired result but I am having trouble.

edges = [('lvl-1', 'lvl-2.1'), ('lvl-1', 'lvl-2.2'), ('lvl-2.1', 'lvl-3.1'), ('lvl-2.1', 2), ('lvl-2.2', 4), ('lvl-2.2', 6), ('lvl-3.1', 'lvl-4.1'), ('lvl-3.1', 5), ('lvl-4.1', 1), ('lvl-4.1', 3), ('input', 'lvl-1')]
graph = nx.OrderedDiGraph()
graph.add_edges_from(edges)
nx.draw(graph, pos=nx.nx_agraph.graphviz_layout(graph, prog="dot"), with_labels=True, node_size=1000, node_color="lightgray")

tree = ete3.Tree()
for parent, children in itertools.groupby(graph.edges(), lambda edge:edge[0]):
    subtree = ete3.Tree(name=parent)
    for child in children:
        subtree.add_child(name=child[1])
    tree.add_child(child=subtree, name=parent)
print(tree) 
#       /-lvl-2.1
#    /-|
#   |   \-lvl-2.2
#   |
#   |   /-lvl-3.1
#   |--|
#   |   \-2
#   |
#   |   /-4
#   |--|
# --|   \-6
#   |
#   |   /-lvl-4.1
#   |--|
#   |   \-5
#   |
#   |   /-1
#   |--|
#   |   \-3
#   |
#    \- /-lvl-1

I've also tried the following but it did not work:

tree = ete3.Tree()
for parent, child in graph.edges():
    if parent not in tree:
        tree.add_child(name=parent)
    subtree = tree.search_nodes(name=parent)[0]
    subtree.add_child(name=child)
print(tree)
#                /-1
#             /-|
#          /-|   \-3
#         |  |
#       /-|   \-5
#      |  |
#    /-|   \-2
#   |  |
#   |  |   /-4
# --|   \-|
#   |      \-6
#   |
#    \- /-lvl-1

# Graph
edges = [('lvl-1', 'lvl-2.1'), ('lvl-1', 'lvl-2.2'), ('lvl-2.1', 'lvl-3.1'), ('lvl-2.1', 2), ('lvl-2.2', 4), ('lvl-2.2', 6), ('lvl-3.1', 'lvl-4.1'), ('lvl-3.1', 5), ('lvl-4.1', 1), ('lvl-4.1', 3), ('input', 'lvl-1')]
G = nx.OrderedDiGraph()
G.add_edges_from(edges)

# Tree
root = "input"
subtrees = {node:ete3.Tree(name=node) for node in G.nodes()}
[*map(lambda edge:subtrees[edge[0]].add_child(subtrees[edge[1]]), G.edges())]
tree = subtrees[root]
print(tree.get_ascii())
#                                /-1
#                         /lvl-4.1
#                  /lvl-3.1      \-3
#                 |      |
#           /lvl-2.1      \-5
#          |      |
# -inputlvl-1      \-2
#          |
#          |       /-4
#           \lvl-2.2
#                  \-6